A Solid of Revolution is obtained by revolving a region in the plane about a line (axis of revolution) in the plane.

The axis either only touches the region or does not intersect the region at all.

Here are some examples.

Slices

Here's the rectangle that we'll revolve about the x-axis
It sweeps out a thin cylinder. If the rectangle has width dx and height h, then the volume of the cylinder is volume = (area of circular base) * dx volume = (Pi * h^2) * dx The horizontal line is the x-axis.

An example of revolving about the x-axis: the Volume of a Sphere

Another Example: revolving y=x^2 about the y-axis

A Third Example: Revolving a region bounded by two curves about the x-axis

y^2 = 4 x

x^2 = 4 y

We are determining the volume of this shape
And not this one
Here they are, side by side. They're different!

Cylinders

2 Cos[Pi/2 - Pi/4 x], 1 <= x <= 3

Take a thin rectangle and revolve it about the y-axis. It sweeps out a thin cylinder.